Pearce-Crump's Dirichlet reciprocal derivative moment conjecture

Let $\chi$ be a primitive Dirichlet character modulo $q>1$, and let $L(s,\chi)$ be its Dirichlet $L$-function, with non-trivial zeros $\rho=\tfrac12+i\gamma$. Assume the Generalised Riemann Hypothesis for $L(s,\chi)$ and that all its non-trivial zeros are simple. Dirichlet reciprocal derivative moment conjecture. As $T\to\infty$,

$$\sum_{0<\gamma\le T}\frac{1}{|L'(\rho,\chi)|^2}\sim\frac{A_q}{2\pi}\,T,$$

and this asymptotic is expected to hold uniformly for $q\le T^A$, for any fixed $A>0$. Here $A_q$ is the arithmetic factor defined in the paper. The paper proves a conductor-uniform lower bound capturing a proportion $\beta/(1+\beta)$ of the conjectured main term, but the full asymptotic is open.